9.1 – Points, Line, Planes and Angles Definitions: A point has no magnitude and no size. A line has no thickness and no width and it extends indefinitely in two directions. A plane is a flat surface that extends infinitely. m A E D

9.1 – Points, Line, Planes and Angles Definitions: A point divides a line into two half-lines, one on each side of the point. A ray is a half-line including an initial point. A line segment includes two endpoints. N E D G F

9.1 – Points, Line, Planes and Angles Summary: Name Figure Symbol Line AB or BA A B AB BA Half-line AB A B AB Half-line BA A B BA Ray AB A B AB Ray BA A B BA Segment AB or Segment BA A B AB BA

9.1 – Points, Line, Planes and Angles Definitions: Parallel lines lie in the same plane and never meet. Two distinct intersecting lines meet at a point. Skew lines do not lie in the same plane and do not meet. Parallel Intersecting Skew

9.1 – Points, Line, Planes and Angles Definitions: Parallel planes never meet. Two distinct intersecting planes meet and form a straight line. Parallel Intersecting

9.1 – Points, Line, Planes and Angles Definitions: An angle is the union of two rays that have a common endpoint. A Side Vertex B 1 Side C An angle can be named using the following methods: – with the letter marking its vertex, B – with the number identifying the angle, 1 – with three letters, ABC. 1) the first letter names a point one side; 2) the second names the vertex; 3) the third names a point on the other side.

9.1 – Points, Line, Planes and Angles Angles are measured by the amount of rotation in degrees. Classification of an angle is based on the degree measure. Measure Name Between 0° and 90° Acute Angle 90° Right Angle Greater than 90° but less than 180° Obtuse Angle 180° Straight Angle

9.1 – Points, Line, Planes and Angles When two lines intersect to form right angles they are called perpendicular. Vertical angles are formed when two lines intersect. A D B E C ABC and DBE are one pair of vertical angles. DBA and EBC are the other pair of vertical angles. Vertical angles have equal measures.

9.1 – Points, Line, Planes and Angles Complementary Angles and Supplementary Angles If the sum of the measures of two acute angles is 90°, the angles are said to be complementary. Each is called the complement of the other. Example: 50° and 40° are complementary angles. If the sum of the measures of two angles is 180°, the angles are said to be supplementary. Each is called the supplement of the other. Example: 50° and 130° are supplementary angles

9.1 – Points, Line, Planes and Angles Find the measure of each marked angle below. (3x + 10)° (5x – 10)° Vertical angels are equal. 3x + 10 = 5x – 10 2x = 20 x = 10 Each angle is 3(10) + 10 = 40°.

9.1 – Points, Line, Planes and Angles Find the measure of each marked angle below. (2x + 45)° (x – 15)° Supplementary angles. 2x + 45 + x – 15 = 180 3x + 30 = 180 3x = 150 x = 50 2(50) + 45 = 145 50 – 15 = 35 35° + 145° = 180

9.1 – Points, Line, Planes and Angles 1 2 Parallel Lines cut by a Transversal line create 8 angles 3 4 5 6 7 8 Alternate interior angles 5 4 Angle measures are equal. (also 3 and 6) 1 Alternate exterior angles Angle measures are equal. 8 (also 2 and 7)

9.1 – Points, Line, Planes and Angles 1 2 3 4 5 6 7 8 Same Side Interior angles 4 6 Angle measures add to 180°. (also 3 and 5) 2 Corresponding angles 6 Angle measures are equal. (also 1 and 5, 3 and 7, 4 and 8)

9.1 – Points, Line, Planes and Angles Find the measure of each marked angle below. (3x – 80)° (x + 70)° Alternate interior angles. x + 70 = x + 70 = 3x – 80 75 + 70 = 2x = 150 145° x = 75

9.1 – Points, Line, Planes and Angles Find the measure of each marked angle below. (4x – 45)° (2x – 21)° Same Side Interior angles. 4x – 45 + 2x – 21 = 180 4(41) – 45 2(41) – 21 6x – 66 = 180 164 – 45 82 – 21 6x = 246 119° 61° x = 41 180 – 119 = 61°